To determine the correct hypotheses to test the owner's claim, let's follow a step-by-step approach:
1. Understand the Owner's Claim:
- The owner claims that the proportion of defective computer chips produced at plant [tex]\( A \)[/tex] is higher than that produced at plant [tex]\( B \)[/tex].
- Mathematically, the owner is suggesting that [tex]\( p_A > p_B \)[/tex].
2. Define the Null and Alternative Hypotheses:
- The null hypothesis [tex]\( H_0 \)[/tex] generally represents an assumption of no effect or no difference. In this case, it should state that there is no difference in the proportion of defective chips between plant [tex]\( A \)[/tex] and plant [tex]\( B \)[/tex].
Therefore, [tex]\( H_0: p_A - p_B = 0 \)[/tex].
- The alternative hypothesis [tex]\( H_a \)[/tex] reflects the owner's claim or the effect we want to test for. Here, it should state that the proportion of defective chips at plant [tex]\( A \)[/tex] is higher than that at plant [tex]\( B \)[/tex].
Thus, [tex]\( H_a: p_A - p_B > 0 \)[/tex].
3. Evaluate the Given Options:
- Option 1: [tex]\( H_0: p_A - p_B = 0 \)[/tex]; [tex]\( H_a: p_A - p_B = 0 \)[/tex]
- This option is incorrect because both the null and alternative hypotheses are the same, which is not valid for hypothesis testing.
- Option 2: [tex]\( H_0: p_A - p_B = 0 \)[/tex]; [tex]\( H_G: p_A - p_B > 0 \)[/tex]
- This option is correct. The null hypothesis states that there is no difference between the proportions ([tex]\( p_A - p_B = 0 \)[/tex]), and the alternative hypothesis asserts that the proportion of defective chips at plant [tex]\( A \)[/tex] is greater than that at plant [tex]\( B \)[/tex] ([tex]\( p_A - p_B > 0 \)[/tex]).
- Option 3: [tex]\( H_0: p_A - p_B = 0 \)[/tex]; [tex]\( H_a: p_A - p_B < 0 \)[/tex]
- This option is incorrect because it suggests that the proportion of defective chips at plant [tex]\( A \)[/tex] is lower than that at plant [tex]\( B \)[/tex], which contradicts the owner's claim.
- Option 4: [tex]\( H_0: p_A - p_E > 0 \)[/tex]; [tex]\( H_Q: p_A - p_E < 0 \)[/tex]
- This option is incorrect because the null hypothesis should typically state that there is no difference between the groups. Additionally, the comparison here is not relevant to the owner's claim and involves notations ([tex]\( p_E \)[/tex] and [tex]\( H_Q \)[/tex]) inconsistent with the problem statement.
4. Final Answer:
- Based on the above evaluation, the correct hypotheses to test the owner's claim are:
[tex]\( H_0: p_A - p_B = 0 \)[/tex] and [tex]\( H_a: p_A - p_B > 0 \)[/tex].
Therefore, the correct set of hypotheses is:
[tex]\[ H_0: p_A - p_B = 0, \quad H_a: p_A - p_B > 0 \][/tex]
